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Cracking Numbers in Envelopes

Q: You are shown two envelopes with numbers written inside them. The numbers are chosen uniformly randomly from the range of 1 to a 100. You are allowed to open one envelope and look at the number inside following which you need to guess if the number in the other envelope is bigger or smaller of the two. You win if your guess is correct. What should your strategy be to have a greater than 50% win probability?

A: This puzzle is similar to the two envelopes paradox. At first pass, it appears that it is impossible to do better than 50% in win probability. But surprisingly, it is possible to do better. What is even more surprising is that it is an amazingly simple solution! All you need to do is to pick a function that is monotonically increasing in the domain 1 to 100. For example, let \(X\) be the number in the envelope you have opened. Let us make a function which takes in the number \(X\) as input and the can be eventually mapped to a range of \([0,1]\) so it "looks" like a probability estimate: A simple one would be to divide \(X\) by a 100.
$$
f(x) = \frac{x}{100}
$$
Next, simply declare the chosen number \(X\) to be the bigger of the two if \(f(X) > 0.5\). This strategy always yields a win rate greater than \(\frac{1}{2}\). Do you see why this works? Lets say the chosen number is 66, you would conclude that you have a \(66\%\) chance that the chosen number is the greater one and your guess would be right \(66\%\) of the time. Likewise, lets choose a function that is not so trivial looking, but continues to be monotonically increasing, for example
$$
f(x) = \frac{x^2}{\pi}
$$
Now map the value returned to a probability as
$$
P(X) = \frac{f(X)}{f(100) - f(1)}
$$
and declare the chosen number to be the greater if \(P(X) > 0.5\). The reason this works is because for a monotonically increasing function \(f(X_1) > f(X_2) \text{ if } X_1 > X_2\), it is as simple as that.

If you are looking to learn probability, some good books to own are shown below

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Linear Algebra (Dover Books on Mathematics)
An excellent book to own if you are looking to get into, or want to understand linear algebra. Please keep in mind that you need to have some basic mathematical background before you can use this book.

Linear Algebra Done Right (Undergraduate Texts in Mathematics)
A great book that exposes the method of proof as it used in Linear Algebra. This book is not for the beginner though. You do need some prior knowledge of the basics at least. It would be a good add-on to an existing course you are doing in Linear Algebra.

Follow @ProbabilityPuzIf you are looking to learn time series analysis, the following are some of the best books in time series analysis.

Introductory Time Series with R (Use R!)
This is good book to get one started on time series. A nice aspect of this book is that it has examples in R and some of the data is part of standard R packages which makes good introductory material for learning the R language too. That said this is not exactly a graduate level book, and some of the data links in the book may not be valid.

Econometrics
A great book if you are in an economics stream or want to get into it. The nice thing in the book is it tries to bring out a oneness in all the methods used. Econ majors need to be up-to speed on the grounding mathematics for time series analysis to use this book. Outside of those prerequisites, this is one of the best books on econometrics and time series analysis.